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5 This question compares a naïve way to take the "sum" and "product" of two sets of integers to the definitions that we actually use in modular arithmetic. (a) Prove that, for any integer m > 0, if X and Y are congruence classes of =m, then the set S = {x+y:x EX‚y≤Y} is a congruence class of Zm, and in fact equals the sum X+Y within Zm. (b) Give an example of an integer m> 0 and two congruence classes X, Y of =m such that the set P= {xy: xe X,y¤Y} is not the product XY within Zm. Write down a general statement about P is related to XY.

5 This question compares a naïve way to take the "sum" and "product" of two sets of integers to the definitions that we actually use in modular arithmetic. (a) Prove that, for any integer m > 0, if X and Y are congruence classes of =m, then the set S = {x+y:x EX‚y≤Y} is a congruence class of Zm, and in fact equals the sum X+Y within Zm. (b) Give an example of an integer m> 0 and two congruence classes X, Y of =m such that the set P= {xy: xe X,y¤Y} is not the product XY within Zm. Write down a general statement about P is related to XY.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.7: Introduction To Coding Theory (optional)
Problem 12E: Suppose that the check digit is computed as described in Example . Prove that transposition errors...
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5 This question compares a naïve way to take the "sum" and "product" of two sets of integers to the definitions that we actually use in modular arithmetic. (a) Prove that, for any integer m > 0, if X and Y are congruence classes of =m, then the set S={x+y:x€X,y€Y} is a congruence class of Zm, and in fact equals the sum X+Y within Zm. (b) Give an example of an integer m > 0 and two congruence classes X, Y of m such that the set P= {xy: xEX, y ≤ Y} is not the product XY within Zm. Write down a general statement about P is related to XY.
Transcribed Image Text:5 This question compares a naïve way to take the "sum" and "product" of two sets of integers to the definitions that we actually use in modular arithmetic. (a) Prove that, for any integer m > 0, if X and Y are congruence classes of =m, then the set S={x+y:x€X,y€Y} is a congruence class of Zm, and in fact equals the sum X+Y within Zm. (b) Give an example of an integer m > 0 and two congruence classes X, Y of m such that the set P= {xy: xEX, y ≤ Y} is not the product XY within Zm. Write down a general statement about P is related to XY.
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